The AAGIGILTV peptide is derived from the Melan-A (Mart-1) protein that is expressed by the majority of fresh melanoma samples and approximately 60% of Melanoma cell lines, as well as normal melanocytes. ((Coulie et al., (1994) J. Exp. Med. 180: (1) 1-4) and Kawakami et al., (1994) PNAS USA 91: 3515) The Class I HLA molecules of these cancerous cells present peptides from this protein, including AAGIGILTV (SEQ ID NO: 43) (Melan-A27-35). The AAGIGILTV-HLA-A*0201 complex appears to be an immuno-dominant target for Melanoma-specific T cells. ((Kawakami et al., (1994) PNAS USA 91: 3515) and (Rivoltini et al., (1995) J Immunol 154: 2257) Therefore, this peptide-HLA complex provides a cancer marker that TCRs can target, for example for the purpose of delivering cytotoxic or immuno-stimulatory agents to the cancer cells. However, for that purpose it would be desirable if the TCR had a higher affinity and/or a slower off-rate for the peptide-HLA complex than native TCRs specific for that complex.